73 



TABLE 3. Wave parameters at minimum critical 



Reynolds number of steady disturbances. 



FIGURE 8. Minimuin critical Reynolds number as function 



of pressure gradient: , steady disturbances, Falkner- 



Skan-Cooke boundary layers with 6 = 45°; , two- 

 dimensional Falkner-Skan boundary layers [from Wazzan 

 et al. (1968) ] . 



(deg) 



FIGURE 9. Effect of flow angle on minimum critical 

 Reynolds number of steady disturbances for 6;^ = 1.0 

 and separation boundary layers. 



dimensional R^,j- is 19,280 compared to R^r = 212 for 

 zero-frequency crossflow instability) . 



The distribution of Rj,j- with 6 is shown in Figure 

 9 for Bh = 1-0 over the complete range of 6 , and 

 for the separation profiles (Bh = -0.1988377) over 

 the range 0° < 9 < 50°. Near 8=0° and 90°, R^r 

 is very sensitive to 9; near, but not precisely at, 

 6 = 45° Rcj. has a minimum. This minimiom occurs 

 close to the maximum of | £ | inf (cf. Table 2), which. 



unlike W„ 



is not symmetrical about 



= 45° 



Table 



3 lists the critical wave parameters for a few com- 

 binations of Bh 3nd 9 . The extensive computations 

 needed to fix these parameters precisely were not 

 carried out in most cases , and so the values in the 

 Table are not exact. The listed ijjgj- was obtained 



j'gi "" 



Boundary Layer with Crossflow Instability Only 



As an example of a boundary layer which is unstable 

 at low Reynolds number only as a result of cross- 

 flow instability, we select Bh = 1-0 ^^^ ^ = 45°, 

 and present results for the complete range of un- 

 stable frequencies . Although this pressure gradient 

 can only occur at an attachment line. Figure 8 leads 

 us to expect that all profiles with a strong favor- 

 able pressure gradient will have similar results. 

 For this type of profile, the minimum critical 

 Reynolds number of the least stable frequency is 

 very close to the R of Figure 7. We therefore 

 choose a Reynolds number well above R^, where the 

 instability is fully developed. 



Figure 10 provides a summary of the stability 

 characteristics at R = 400. For a given frequency, 

 the eignevalue o {i>) can be computed as a function 

 of either k or 4/, with the other parameter given as 

 the second eigenvalue. For strictly crossflow 

 instability, k is the more suitable independent 

 variable as i|) can have an extremum in the unstable 

 region. All unstable eigenvalues of a given fre- 

 quency with a specified increment in k were calcu- 



